# Subspaces and Rank-Nullity

After finishing a chapter on Rank-Nullity in linear algebra I came across the following problem:

Suppose that $M,N$, are subspaces of a finite-dimensional vector space $V$ such that the dimensions of

$(M+N)/N = 2$,

$(M+N)/M = 3$,

$(M \cap N) = 4$

Then what are the dimensions of $N$, $M$, and $M+N$.

Thoughts:

$\dim (M+N)/N = \dim (M+N) - \dim (N) = 2$, and

$\dim (M+N)/M = \dim (M+N) - \dim (M) = 3$, Thus $\dim (N) - \dim(M) = 1$ . This is where I am stuck. Hints appreciated.

## 1 Answer

Use the formula for the sum of two vector spaces.

For reference, it is in the question of this: Dimension of the sum of two vector subspaces